Nilpotent approximations of control systems

We show how to constructively approximate an affine control system x = X0(x) + u1 (t)X1(x) + ... + uk(t)Xk(x) on IRn by a control system of this same form, and also on IRn, for which the describing vector fields generate a nilpotent Lie algebra, or at least an important subalgebra of the algebra they generate, is nilpotent.